Tauri's RustWebChromeClient overwrites any chrome-client override after onWebViewCreate returns, so onProgressChanged-based injection never fires. Switch to WebViewCompat.addDocumentStartJavaScript (the same API Tauri uses for its own init scripts) — registered on the WebView object, survives chrome-client swaps, runs before page JS on every navigation. Falls back to direct evaluateJavascript on the (rare) devices that lack DOCUMENT_START_SCRIPT feature support. Reads insets synchronously in onWebViewCreate so the first registration has real values even if the insets listener hasn't fired yet. Fixes Android header overlap behind the status bar.